Problem 14 - Algebra 2

Question

Factor each expression. $$ x^{2}-13 x+12 $$

Step-by-Step Solution

Answer

\((x - 1)(x - 12)\)

Step 1: Identify a, b, and c in the quadratic expression

Recognize that the quadratic expression is in the form of a standard quadratic trinomial, which is written as: \(ax^2 + bx + c\). In the given expression \(x^2 - 13x + 12\), we identify \(a = 1\), \(b = -13\), and \(c = 12\).

Step 2: Find two numbers that multiply to ac and add to b

Look for two numbers that multiply to get \(a \times c = 1 \times 12 = 12\) and add up to \(b = -13\). The two numbers that satisfy these conditions are -1 and -12.

Step 3: Rewrite the quadratic expression using the two numbers

Rewrite the middle term of the quadratic expression using the two numbers found in the previous step. The expression \(x^2 - 13x + 12\) can be written as \(x^2 - x - 12x + 12\).

Step 4: Factor by grouping

Group the terms into two pairs and factor out the common factors from each pair. The pairs are \((x^2 - x)\) and \((-12x + 12)\). Factoring out the common factor from each pair gives us \(x(x - 1)\) and \(-12(x - 1)\).

Step 5: Factor out the common binomial factor

The common binomial factor in both groups is \((x - 1)\). Factor this out to get the final factored form of the expression: \((x - 1)(x - 12)\).

Key Concepts

Quadratic Trinomials

A quadratic trinomial is a polynomial with three terms that can be written in the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and the highest power of the variable is two. This structure forms the backbone of many algebraic processes, including factoring and solving quadratic equations.

To master factoring a quadratic trinomial, it's essential to identify the values of \(a\), \(b\), and \(c\). They represent the coefficients of the squared term, the linear term, and the constant term, respectively. For instance, in the expression \(x^2 - 13x + 12\), \(a=1\), \(b=-13\), and \(c=12\).

The goal of factoring is to break down the trinomial into a product of two binomials. This process can become more intuitive with practice. Recognizing patterns, such as the difference of squares or perfect square trinomials, can also aid in factoring more complex expressions efficiently.

Factor by Grouping

Factoring by grouping is a method that involves rearranging and grouping terms in a polynomial to find common factors. This technique is particularly useful when dealing with four-term polynomials or when the 'a' coefficient in a quadratic trinomial is not equal to one.

The method starts by splitting the middle term into two terms that can be grouped with the first and last terms. The grouping should be done in such a way that each group has a common factor that can be factored out. After grouping the terms, factor out the common factors for each group to simplify the expression.

For instance, the expression \(x^2 - x - 12x + 12\) can be grouped into \((x^2 - x)\) and \( (-12x + 12)\). Factoring out \(x\) from the first group and -12 from the second group gives us \(x(x - 1)\) and \( -12(x - 1)\). The next step is to identify the common binomial factor between these two terms, leading to complete factorization of the expression.

Binomial Factor

The binomial factor refers to the two-part expression that is common across the terms resulting from the factor by grouping method. This shared binomial is what's left after factoring out the common factors from grouped terms. The presence of a common binomial factor is the key to successfully factoring the original quadratic trinomial.

In the example \(x(x - 1) - 12(x - 1)\), \(x - 1\) is the binomial factor. Finding this commonality allows you to pull \(x - 1\) out of both terms, leaving you with \(x - 1\) and \(x - 12\) as the binomials that, when multiplied together, give back the original quadratic trinomial. This results in the factored form \( (x - 1)(x - 12)\), demonstrating that the expression can indeed be decomposed into more straightforward, multiplicative components.